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Update 1: Lezales Got Swole
From here on, updates will assume a basic familiarity with the battle mechanics of Final Fantasy Tactics.
Story: Basically, the story starts in the chapel of Orbonne Monastery. Princess Ovelia is praying at the altar while her bodyguard and attendant (Agrias and Simon, respectively) wait for her to finish. Gafgarion and his band of mercenaries (who have been hired to escort Ovelia) barge in and ask why she's taking so long. Ovelia wraps it up and gets ready to leave when a knight stumbles into the altar with a stab wound, saying that Goltana's men have showed up. Everyone goes outside and gets ready to stab the shit out of some dudes.
And finally, we can start our first fight!
Agrias: That's just what Goltana wants us to do! Just let them go!
Gafgarion: That's impossible!
So then the fight actually gets underway.
And Gafgarion breaks out Explosion, one of the spells in the completely revamped Black Magic skillset. He will do this frustratingly often, because the AI tends to resort to whatever skill will do the most damage. It's mostly annoying because of the charge time.
The way charging spells or attacks works is that the game inflicts "Charging" as a status on the character who uses those abilities. When the spell resolves, the game checks to see if all of the conditions are met (right weapon equipped for panel selected, enough MP, target still alive if spell is based on target, etc.) and then performs the spell. One of those checks is if enough clockticks have passed from the time the spell was selected to the time that charging ends. If charging ends prematurely, then the spell never goes off. Stunning Strike (one of the ArcherMarksman's new abilities) cancels out the spell by cancelling the "Charge" status early. Knowing that the Marksmen on the map possess this ability will not deter Gafgarion from trying to cast spells every turn if he thinks they'll do more damage. 
Our hero. The astute among you will note that his Brave/Faith levels are set to a random value, rather than 70/70 like they are at the beginning of Chapter 1. This doesn't affect anything, but it's something notable.
I will be showing the entire lineup of enemies and the setups of my allies before every battle, so with that in mind, here's your lineup:
I'm pretty sure the skillsets are random even for this first fight. This can lead to big problems when you only have control over one character.
Agrias is still pretty sturdy. The Yin-Yang magic is especially useful, though it doesn't really make use of her exceptionally high magic stat.
Ramza is pretty decked out. It's unfortunate that you don't get to keep this stuff.
Sneak peak of the new abilities of the Squire. I'll get into more detail with this when we go into the job breakdowns.
Lezales. He's built like a brick shithouse. Note his R/S/M abilities: they can ruin you. The first time I played FFT 1.3, Lezales had Item as his secondary ability, and he had Hi-Potion. I actually lost the first fight in Final Fantasy Tactics. I was devastated.
Anyways, he'd be more manageable if you had control over all of the characters, but as it stands, he's still pretty easy to beat. If he gets lucky with Weapon Break (which he had here), things will get ugly in a hurry.
Rad. Rad is useless, and we will not see Rad for a while after this fight. Rad will serve his team by distracting the enemy Marksmen for three turns until he dies of arrow wounds.
Alicia and Lavian. Pretty interchangeable, except Lavian has the Item ability, which would be useful if I were controlling people, but instead means that she will spend the majority of the fight trying to revive Rad.
Wezaleff and Viggs. Wezaleff has Stunning Strike and Viggs has Item/Phoenix Down, so both of them are a pain in the ass and have to be dealt with as soon as possible.
I don't think he got the opportunity to use White Magic in this fight, but as resident medic, he's a high priority target. Most fights in 1.3 revolve around crippling the enemy's ability to recover from damage, and a disproportionate number of enemies have the ability to regenerate either themselves or their teammates. This extends fights dramatically, and yes, it's really, really fucking annoying.
Finally, Dish. Pretty unimportant, but if he starts Power BreakRuining your teammates, it can turn this fight into a two-hour long slugfest. The RNG will fuck you every time, too.
Throw Stone costs less JP than before and remains somewhat useful for a while. It multiplies your PA by a number between 1 and 3, and it has 100% knockback, which is useful against enemies that stand on rooftops. Fall damage doesn't trigger reaction abilities and can't be avoided, so it's about as safe as damage can get in this game.
This is Power Ruin. It's got a range of 3 and costs 6 MP.
Works just like Power Break, except you can't use Two Swords to totally cripple enemies anymore. It does improve the utility of Knights, though.
Overall, this fight goes about how you remember it from the original FFT.
This is Explosion. It has a 25% chance of causing Oil, which actually works now.
By the way, Holy Sword (and all Swordskills, including All Swordskills :v
has been nerfed. It costs MP to use...
...and is subject to physical evasion.
Night Sword is still fantastic, though less so than before.
Dash still has 100% accuracy and doesn't trigger some reaction abilities, so it's still the method of choice for dispatching enemies with critical HP. It also has a 25% chance of cancelling charging, so if you're desperate it's a pretty good move. Rad did not live to see the sun rise. I'll go over the rest of the abilities for Squires in a separate update.
Agrias using Spell Absorb so she can go back to dropping magical sword crystals on things. Gafgarion is using Dark Sword, which is only usable by one other character in the game and is very useful against boss characters.
Gafgarion casts another Explosion, which results in Oil. The idea behind Explosion is that you can chain cast it for heavy damage if you get lucky with the status effect triggering.
How nice, Gaf is going to cure him of his oilyness. 
At this point, the fight is over. Lezales is just being a dick at this point. Three enemies earlier, I would've been pretty fucked.
Eventually, my team finishes beating the shit out of him.
So, a woman screams and then Agrias remembers that she was supposed to be protecting the princess and instead left him with a feeble old man. Ovelia is captured by a devilishly handsome knight named Delita, who then rides off on a mutant land bird as Ramza reminisces about the good old days of murdering bandits with his childhood friend.